The close interplay between antiferromagnetism (AF) and superconductivity (SC) has been of great interest both to experimentalists and theoretical physicists for many years . The competition of two different order parameters has been observed in various systems such as organic superconductors , ternary compounds , heavy fermion compounds , as well as high temperature superconductors in the latest . In this Thesis , the interplay between antiferromagnetism and d -wave superconductivity is studied in a mean-field approximation for a generic microscopic Hamiltonian with short-range repulsion and nearest neighbor attraction . In the square lattice it is showed that the transition between superconductivity and antiferromagnetism occurs through a phase , where both order parameters coexist with a third one , which is dynamically generated , with spin-triplet amplitude . This dynamical generation of a new order parameter is not restricted to a system with antiferromagnetism and d -wave superconductivity , but is a generic feature for fermionic systems . By neglecting the third parameter , antiferromagnetic and superconducting states are studied in the honeycomb lattice . We also studied the competition between s -wave , extended s -wave , -wave , and d xy -wave uperconducting phases in half-filling . The interest in strongly correlated systems in frustrated lattices has increased recently because of the possible realization of exotic magnetic states , spin and charge separation in two dimensions , and the discovery of superconductivity in $ Na_xCoO_2 . yH_2O$ . Many researchers have discussed superconductivity in non-Bravais lattices , mainly using selfconsistent spin fluctuation approaches to the problem . The honeycomb lattice , which is made of two interpenetrating triangular lattices , has received special attention after the discovery of superconductivity in MgB_2$ . Additionally , the honeycomb lattice has been shown to stage many different types of exotic physical behaviors in magnetism and the growing experimental evidence of non-Fermi liquid behavior in graphite has led to the study of electron-electron correlations and quasiparticle lifetimes in graphite . As the next step , We discussed the magnetic phases of the Hubbard model for the honeycomb lattice in two dimension by Green function . A ground state phase diagram is obtained depending on the interaction strength U and electronic density n . The magnetic excitations (spin waves) in the antiferromagnetic insulating phase are calculated from the random-phase approximation for the spin susceptibility . By using the Dyson equation for spin susceptibility , the critical U , in which transition from paramagnetic phase to ferromagnetic and antiferromagnetic phases occurs , is obtained . We also study theoretically the possibility of s -wave superconductivity in honeycomb lattice within the Hubbard model . The superconducting order parameter is defined by introducing the anomalous Green function . we could obtain the relation of superconducting order parameter with respect to electronic density . Keywords: Honeycomb lattice , Phase transition , Antiferromagnetism , Superconductivity , Mean field theory , Green function.